I feel confused about:
1:It says that "otherwise H contains an infinite cyclic characteristic subgroup C," however, by definition, if H is elementary, we can only get that H contains an infinite cyclic subgroup with finite index, why does that it imply H has an infinite cyclic characteristic subgroup?
2: Why $C_G$(C) has finite index in G?
3 Why C has finite index in $C_G(C)$?
I can answer your questions 1 and 2. For 1, suppose that $H$ contains the cyclic subgroup $D$ of finite index. By replacing $D$ by its core in $H$ (i.e. the intersection of its conjugates in $H$), we may assume that $D\unlhd H$. Let $|H:D|=n$. Then $h^n \in D$ for all $h \in H$, and the subgroup $C:= \langle h^n \mid h \in H \rangle$ is characteristic in $H$ and is contained in $D$ with index at most $n$, so it is infinite cyclic.
Question 2 follows from the fact that ${\rm Aut}(C)$ is finite - in fact it has order $2$ and $G/C_G(C)$ is isomorphic to a subgroup of ${\rm Aut}(C)$, so $|G/C_G(C)| \le 2$.
I am not familiar with all of the terminology. What is a special element? What does $E_G(H)$ mean? What does loxodromic mean?